A construction of Boolean algebras from first-order structures

Annals of Pure and Applied Logic 59 (1993) 239-256

North-Holland

239

A construction of Boolean algebras from first-order structures

Sabine Koppelberg 2. Math. Institut der FU, Arnimallee 3, W-1000 Berlin, 33, Germany

Dedicated to J.D. Monk on the occasion of his 60th birthday

Communicated by A. Prestel

Received 27 June 1991

Abstract

Koppelberg, S., A construction of Boolean algebras from first-order structures, Annals of Pure

and Applied Logic 59 (1993) 239-256.

We give a construction assigning classes of Boolean algebras to first-order theories; several

classes of Boolean algebras considered previously in the literature can be thus obtained. In

particular it turns out that the class of semigroup algebras can be defined in this way, in fact by

a Horn theory, and it is the largest class of Boolean algebras defined by a Horn theory.

We present, in this paper, a construction which assigns a Boolean algebra

B(d, 2) to an -Y-structure Sp and a theory 2 in a language .9(P) z 5?:

The interesting aspect of the construction is that some well-known (and some not

so well-known) classes of Boolean algebras are model-theoretically defined, i.e.,

representable in the form

for some theory T in 9.

The construction is given in Section 1 of the paper. In Section 2, we give a list

of examples of model-theoretically defined classes. One of these, the class of

semigroup algebras, is considered in Section 3: it is the greatest model-

theoretically defined class K(T, E) with 2 essentially a Horn theory, and its

members have quite finitary properties. In Section 4, we make some observations

how properties of T imply properties of the algebras in K(7’, 2).

Correspondence to: S. Koppelberg, 2. Math. Institut der FU, Arnimallee 3, W-1000 Berlin 33,

Germany.

0168~0072/93/$06.00 0 1993 -Elsevier Science Publishers B.V. All rights reserved

240 S. Koppelberg

For unexplained notation and results in model theory, the reader might consult

[6], in Boolean algebras [12]. In particular, the operations and constants in

Boolean algebras are denoted by + , *, - , 0, 1, the Boolean partial ordering by s.

I should like to thank Qi Feng, Sakae Fuchino, and Steve Simpson for asking

stimulating questions and Lutz Heindorf for making me familiar with semigroup

algebras and indicating some improvements.

1. The construction

Let 3 be a language for first-order predicate logic, P a unary predicate not in

9, and Z’(P) = 2 U {P}. Call a sentence o in Z(P) universal over 3 if

where q(Z) is a Boolean combination of formulas a(X) which are in 3 or of the

form P(t(f)), t(i) a term in 3. (Thus o is universal if .Z’-formulas are thought of

as being atomic.) Assume 2 is a fixed theory in Z(P) such that each o E ,Y is

universal over 3.

Here is our favorite example: 3 = { +, ., 0, l} is a language for fields, and the

axioms of 2 say that P is (the set of positive elements of) an ordering. More

exactly, _Z = { ol, . . . , a,} where

0,: Vx [P(x)+x # 01,

a,: vx [P(x) v P(-x) v x = 01,

03: v_x VY [P(x) A P(Y)-+P(X +y)l,

04; v_x VY [P(x) * P(Y>--+P(X -Y)l;

so the axioms of Z are universal over 5’.

For an arbitrary set A, identify the power set B(A) of A with A2 = {x: x is a

function from A to 2 = (0, l}}, via characteristic functions. Giving 2 the discrete

topology and A2 the product topology, we obtain the natural topology on C?‘(A) which makes ??‘(A) a Boolean (i.e., compact, Hausdorff, zero-dimensional) space.

A subbase for this topology consists of the sets

{UGA: a E U}, {UsA:a$U}

where a varies through A. In particular, we can do so if A is the underlying set of an Z’-structure

&=(A,. . .). Given the theory _Z (universal over 3) as above, call U E A a

E-set, or a Z-subset of ti if the .5!?(P)-structure (ti, U) (in which P is interpreted

by U) satisfies 2. Then define:

X(ti, Z)= {U&A: U a Z-set}

= {UcA: (~4, U)kZ},

Boolean algebras from first-order structures 241

a subspace of P(A),

B(s~, E) = the Boolean algebra of clopen subsets of X(A!, ,Y),

and, for T an arbitrary Y-theory,

K(T,Z)={B(SI,CZ):&~T},

a class of Boolean algebras.

In our example above, X(&, 2) is, for an arbitrary field .aQ, the space of all

orderings of &. It is well-known among field theorists, and also shown by the

following proposition, that this is a Boolean space.

1.1. Proposition. For J?Z universal over 3, X(&, .Y) is a closed subspace of P?(A), hence Boolean.

Proof. Write X = X(&, 2); we show that 9(A)\X is open. So let I/ E 9(A) \X; we find a neighbourhood N of U in B(A) disjoint from X. Since U 4 X, fix o E _Y

such that (a, I/) l# o. Say u = Vx q(X) as above, so fix a sequence 5 in A such that

(4 U) # dd. c onsider the following finite subsets of A:

u = {t”[ti]: t(Z) an Z-term, P(t@)) a subformula of 97, t”[ti] E U},

v = {~“[a]: t(x) an Z-term, P(t(Z)) a subformula of q, t”[ti] 4 U}. Then

N={VEA:~GV,VGA\V}

is a neighbourhood of U disjoint from X. Cl

It is, of course, possible to define the Boolean algebra B(&, 2:) from ~4 and 2

in a purely algebraic resp. model-theoretic manner; this is carried out below.

However, it seems to be a matter of taste and convenience whether to consider

the space X(&, E) or the Boolean algebra B(&, 2) as the object of primary

interest - e.g. in the above example (orderings of fields), field theorists consider it

more natural to study the space of orderings rather than its dual algebra.

In the next lemma, recall that for & = (A, . . .) an Z-structure and M G A, 3, = 3 U {a: a E M} is the language with a new constant symbol a for each

a EM, Se, = (4 a)asM is the expansion of & to an .Y,-structure in which a,

a E M, is interpreted by a, and eldiag L& = Th(&*) is the elementary diagram of

~4. Let Fn(A, 2) = { e: e maps a finite subset of A into 2 = (0, l}}. Given

a=(A,. . .) and 2 (universal over ,ie), call e E Fn(A, 2) consistent (with .& and

2) if there exists a E-subsets U of A such that the characteristic function

x,-, :A+ 2 of U extends e, i.e., for a E dom e, e(a) = 1 implies a E U and e(a) = 0 implies a $ U.

1.2. Lemma. (a) If .d is an elementary submodel of 93 and V is a Z-subset of 9, then V fI A is a Z-subset of ti.

242 S. Koppelberg

(b) For e a finite partial function from a subset of A to 2, the following are

equivalent: (i) e is consistent with ~4, 2;

(ii) there are 53 > ~4 and a z-subset V of 93 such that e c xv;

(iii) the 5&(P)-theory

efdiag(&) U 2 U {P(g): a E dom e, e(a) = 1)

U {+(_a): a E dom e, e(a) = 0}

is consistent. (c) Zf ti < 53 and U is a E-subset of &, then there is a ,I?:-subset V of 95’ such that

U=VnA.

Proof. (a) holds because 2 is universal over 2, and (b) follows from (a). (c)

follows from (b) and the compactness theorem of first-order logic. 0

If all o E JY happen to be universal closures of quantifier-free formulas of z(P),

then in (a) and (b) of the lemma we can replace the condition 6% z ti by %’ 2 &,

and the elementary diagram eldiag ti of d by the quantifier-free diagram diag s4. A bit of Stone duality shows the following, using 1.2.

1.3. Proposition. (a) The Boolean algebras B(&, 2) is isomorphic to Fr(A)/Z where Fr(A) is the free Boolean algebra independently generated by A (thus A s Fr(A)) and Z is the ideal of Fr(A) generated by

J = {p,: e E Fn(A, 2) inconsistent with Oe, E};

here pe is the elementary product

pe = KI e(a)a aedomp

over A (cf. [12, 4.31 for notation).

(b) B(4 2) is, up to isomorphism, the Lindenbaum-Tarski algebra of quantifier-free sentences in the language {P} U {a: a E A}, modulo the theory ,Y U eZdiag(s4). 0

One consequence of this proposition is that the assignment d, Z~H B(s4, 2) is

absolute, for transitive models M c N of ZFC set theory. For assume J& and 2 are

elements of M. Then X”(&, x) may be a proper subset of XN(&, 2) (in fact,

X”(&, z) = XN(&, 2) fl M), but the algebra Fr(A) and its subsets J and Z are

absolute for M s N. Moreover, the degree of effectiveness of the constructions &, LY++ B(&, 2) can

easily be described. E.g. assume .L!? is finite, 2 is recursively enumerable, ti is a

countable structure and eldiag ti (or, if each o E 2 is the universal closure of a

quantifier-free formula, diag a) is recursive. Then B(sZ, E) is a recursively


enumerable Boolean algebra in the sense of [17], because J and I in 1.3(a) are

recursively enumerable. In fact, in many cases, B(&, 2) will be a recursive

Boolean algebra in the sense of [17]. E.g. let J& = (A, <) be a recursive linear

order and let 2 be as in Example 2.9; B(&, Z) is the interval algebra of &?. Now

J in 1.3(a) is recursive since it is decidable from diug & whether e E Fn(A, 2) is

consistent with &, Z: e is consistent iff, in the linear order &, max{u E A: e(u) = 1) < min{a EA: e(a) = O}. The ideal I in 1.3(a) is recursive since, for a =pe, + . . . +pe, E Z+(A) written in additive normal form over A, a E Z iff all pc, E J.

2. Examples

Let us say that a class K of Boolean algebras is model-theoretically dejined if

K = lK(T, _Z), for some T and Z (universal over 2). We give, in this section, a list

of more or less well-known classes which are model-theoretically defined. In the

first examples of our list, the topological approach (construct first X(&, .Z), then

B(d, 2)) seems quite natural or gives, in fact, the original definition of the class

under consideration.

2.1. The class of all Boolean algebras. Let _Y = { +, e, -, 0, l} be the language of

Boolean algebras, T any first-order axiomatization of the theory of Boolean

algebras. Let Z say that P is (i.e., defines) an ultrafilter, i.e., ,Z consists of the

sentences

01: P(l),

a,: lP(O),

a,: v.x b [P(x) A P(y)+ P(x * y)],

a,: vxvy[P(x)Ax~y=x~P(y)],

a,: vx [P(x) v P(-x)].

This 2 is universal over 3. Given Se k T, i.e., & a Boolean algebra, X(&, 2) is

the space of all ultrafilters with the Stone topology; by Stone duality, B(&, 2:) =

Clop(X(d, 2)) = L&?. Thus K(T, E) is (up to identification of isomorphic al-

gebras) the class of all Boolean algebras.

2.2. Once more the class of all Boolean algebras. We consider the ‘favorite

example’ of Section 1: 2 is the language of fields, T the theory of fields, t: says

that P is an ordering. A non-trivial theorem by Craven (cf. [7]) states that

K(T, _Y) is the class of all Boolean algebras.

2.3. Graph algebras. Here .P? = {r} is the language of graphs, r a binary relation

symbol. T is the theory of (undirected) graphs (stating that r defines a symmetric

244 S. Koppelberg

relation). _Z says that P is a complete subgraph:

VCC VY [P(x) A P(Y 1 A x f Y + +, Y)l.

For a graph &, X(&, 2) is the graph space of &, as defined by Bell [l]. We

might call B(&, 2) the graph algebra of &.

2.4. Free algebras. Let 3 and T be arbitary (e.g., .3?= T=O) and let Z=O. Thus

for & k T, X(.d, 2) = 9’(A) = A2, and B(&, 2) is the free Boolean algebra over

IAl free generators. If T has a model of size K, for each cardinal K, then W( T, 2) is the class of all free Boolean algebras.

2.5. Finite-cofinite algebras. Again, let 3’ and T be arbitrary; let 2 have the

single axiom

vx Vy [P(x) A P(y)+ x = y].

Then X( &, 2) = {U G A: 1171 s l}. This is the discrete space with IAl + 1 points if

A is finite and the one-point compactification of a discrete space with JAI points

otherwise; in the latter case, the empty subset of A is the only non-isolated point

of X(&, 2). It follows that B(&, 2) is the finite-cofinite algebra over a set of

cardinality IA( + 1.

2.6. Exponential algebras. Let us recall from topology that, for a topological

space X, exp(X), (the exponential of X) is the set of all non-empty closed

subspaces of X, with the topology having the base

{s(ur, . . . , u,): ul, . . . , u, LX open}

where

s(u,, . . . 9 u,)={YEexp(X):Ysu,U...Uu,,

Yflu,#0,. . .) Yflu,#O}.

If X is Boolean, then so is exp(X), and

{s(ui, . . . , u,): ul, . . . , u, GX clopen}

is a clopen base of exp(X). For a Boolean algebra & and X = Ult ~4 its dual

space, we might call the algebra exp(d) dual to exp(X) the exponential algebra

of a; this notion has been studied by Heindorf [9]. We show that the class

M = {exp(d): d is a Boolean algebra)

is model-theoretically definable. For, let 3 be the language and T the theory of

Boolean algebras, as in Example 2.1; let _Z say that P defines a proper filter, i.e.,

z= {q, . . . , 04} ; the oi being defined in 2.1. Then K = W( T, 2) since, for ti a

Boolean algebra with dual space X, we prove that X(&, 2) is homeomorphic to

exp(X). To this end, let s :A+ Clap(X) be the Stone map with s(a) = {x E

Boolean algebras from jirst-order structures 245

X: a EX} and consider the map

q:X(4 2)-+exp(X),

v(F) = fFs(a).

q(F) is the closed subset of X dual to F (see [12, 7.251); in particular, by Stone

duality, q is a bijection. It is easily shown that q is continuous.

We proceed to demonstrate how some more classes 06 of Boolean algebras are

model-theoretically defined. In these examples, K will be a class of algebras

generated by some subset with a special property. We will apply the following

observation from Stone duality. Suppose B is a Boolean algebra generated by a

subset H. The function

f : Ult B -+ P(H),

X -xflH

is then a homeomorphism from the Stone space Uft B of B onto a closed subspace

of P(H) where P(H) is given the natural topology- see Section 1. It follows

from Sikorski’s extension criterion (see [12, 5.51) that V E H is in ranf iff V

satisfies

(I) if a,, . . . , a,EV, b ,,..., b,EHand

a,. a.- .a, sb,+--.+b,, then some bj is in V.

Note that, for m = 0 resp. n = 0 this implies:

if a,, . . . , a,EV,thena,.....a,#O;

if b,, . . , b,EHandb,+.. . + b, = 1, then some b, is in V.

2.7. Tail algebras. Given a partial order .FZ = (A, =s), define for t E A,

b,={xEA:t%x},

the tail of A generated by t,

and let B = tailag(d), the tail algebra of &, be the subalgebra of the power set

algebra of A generated by H. This notion is due to Brenner (unpublished); see

also [14]. Note that the assignment t-b, is one-one and order-reversing.

A subset V of H, say V = {b,: t E U} where U E A, corresponds to an ultrafilter

of B iff (by (1)) U satisfies

(2) if y,, . . . , YnEU, Zl,..., z, E A and b,, fl . - . r-16,” G b,, U . . . U b =,,,,

then some 2; E U.

246 S. Koppelberg

The relation byI ~3 - - - n by, c b,, U . . . U b,_ can be expressed by the formula

Q&&j, 2); Vx [yl <x A * - * r\y, =sx+zI Gx v.. . v 2, sx].

Hence

Uft(ruilulg(se)) =X(.& Z),

where 2 = {s} and

z = {& . * a vyH VZ,. * * vZ,n [P(yl) A * * * A P(yn) A Q&,,u, 2)

+ P(q) v * . . v P(zm): n, m E w}.

Thus, for T the theory of partial orders, K(T, lY) is the class of tail algebras.

2.8. Pseudo-tree algebras. A partial order & = (A, G) is called a pseudo-tree if,

for each t E A, the initial segment {x E A: x s t} is totally ordered. (Thus trees are

well-founded pseudo-trees.) We then call the tail algebra of & the pseudo-tree algebra of & and denote it by treealg(.@; it was defined and studied for trees by

Brenner (cf. [3,4]) and for pseudo-trees by Monk and the author in [14]. Letting

2 = {c}, T the theory of pseudo-trees and .Z as in 2.7, we find that K(T, 2’) is

the class of pseudo-tree algebras.

In fact, Z can be simplified here: in a pseudo-tree ~2 = (A, s), call I/ GA an

initial chain if it is a chain in &, an initial segment of &, and, if the set M of

minimal elements of & happens to be finite and every element of A lies above

some element of M, then U fl M # 0. It is easily checked that U E A is a Z-set (2

as in 2.7) iff it is an initial chain. Hence also K(T, 2’) is the class of pseudo-tree

algebras, where 2” has the axioms

v.X VJ’ [P(X) A P(J’)+X <y V y CX],

VX vy [p(X) A J’ s X-+ p(Y)],

vx, . . ~vx,[vx(x,~xv-~~vx,~x)

4 P(xl) v . . . v P(xn)], n E 0.

2.9. Interval algebras. We consider here the special case of 2.8 in which

ti = (A, G) is a linear order. tuilalg(&) = treealg(&) is then called the interval algebra of & and denoted by intalg sA. Its elements are finite unions of half-open

intervals [xi, y,) U . . - U [x,, yn) where --m <x1 <yl < * - * <x,, < y, G +a in &;

its ultrafilters correspond to initial segments of &. Still simplifying 2:’ of 2.8 a bit,

we see that the class of interval algebras is M(T, 2”) where Y = {s}, T is the

theory of linear orders, and 2” has the axioms

v.X V [P(x) A y =zx+ P(Y)], vz [Vx (z cx)+ P(z)].

2.10. Semigroup algebras. Heindorf defines in [9] a Boolean algebra B to be a

semigroup algebra if there is a subset H of B generating B such that H is closed


under the Boolean multiplication . (hence a semigroup, under .), 1 E H, 0 E H,

and H\(O) is disjunctive, i.e., for a and bI,. . . ,b,~H\{0}, a~b,+..-+b,

implies that a =Z bj, for some j. Consider the embedding f : Ult B + 9”(H) defined

before 2.7; by disjunctiveness of H\(O), a subset V of H is in ranf iff V is a

proper filter in the semilattice (H, -). It follows that the class of semigroup

algebras coincides with K(T, 2) where 2 = {., 0, l}, T says that ti = (H, -, 0, 1)

is a commutative idempotent semigroup (i.e., a semilattice) with 0 and 1, and _Z

says that P defines a proper filter; i.e., with the sentences ui defined as in 2.1,

E= {a,, . . ) Q}.

The following diagram shows the inclusions known, up to now, between the

above-mentioned classes.

2.4 - 2.3 - 2.10 - 2.7 - 2.1.2.2

2.9 - 2.8 /

/ 2.5

Here arrows mean inclusion, e.g., 2.9 +2.8 says that the class mentioned in 2.9

(interval algebras) is included in the class mentioned in 2.8 (pseudo-tree

algebras). Among the non-obvious inclusions, 2.84 2.10+ 2.7 is proved in [14];

2.6+ 2.10 and 2.3-, 2.10 follow from 3.3 in Section 3. 2.6-+ 2.10 was first stated

in [9].

3. Semigroup algebras and Horn theories

We point out here the prominent role of the class S of semigroup algebras (see

2.10) among the model-theoretically defined classes. See e.g. the position of s in

the diagram in Section 2. We shall quote in this section some theorems on

semigroup algebras and comment on their meaning in model-theoretically defined

classes.

The following result is a consequence of the duality between discrete and

compact zero-dimensional semilattices as presented in [ 111.

3.1. Theorem (Hofmann, Mislove, Stralka; Heindorf). A Boolean algebra is a

semigroup algebra iff, on its dual space X, there is a continuous multiplication

. :X x X-X which makes X into a topological semilattice (i.e., . is associative,

idempotent, commutative).

248 S. Koppelberg

Let the theory 2 be universal over 9, as defined in Section 1. Up to logical

equivalence, we can (and will now) assume that

1. for u = V%! q(X) in 2 and Z’(t(f)) an atomic subformula of Q?, t(z) is a

variable;

2. ~(2) is a disjunction of exactly one .5&-formula and formulas of the form

P(x,) resp. +(x,), xi a variable.

Call cp(i) as in 2. a basic Horn formula over 27 if

q=$J,v...vr/J,

where ~ZJ~ is an ._Y-formula, at most one qk has the form P(x,), and the other ones

have the form iP(xi). I.e., up to renaming of variables, Q, has the form

(3) ii ZW * 44-t Z%+J (a in 9)

or

(4) z&l P(xi)+P(9 (P in 2).

Call 2 a Horn theory over .2X if each o E 2 is the universal quantification of a

basic Horn formula over 2. Call a class K of Boolean algebra Horn dejinable if

K = K(T, 2) for some Horn theory J? over 55’.

3.2. Remark and definition. Assume 2 is a Horn theory over 5’ and ~4 is an

.J!?-structure. If Z # 0 and, for i E I, Ui is a z-subset of A, then niel lJi is a z-set.

Thus if M GA is such that there exists a z-set including M, then

cl(M) = n {U E X(4 2): M G U},

the closure of M, is the least x-set including M.

3.3. Theorem. The class S of semigroup algebras is Horn-definable; it is, in fact, the greatest Horn definable class of Boolean algebras.

Proof. It was shown in 2.10 that !5 = lK(T, E) where 2 = { oi, . . . , o,}, the oi as

in 2.1. This _E is a Horn theory.

Now assume K = K(T, 2) is a Horn-definable class. For a k T, 3.2 shows that

the space X = X(Se, 2) is closed under the multiplication

U,VHlJ.V=Ur)V;

(X, .) is a semilattice with cZ(0) as its least element, and this multiplication is

obviously a continuous operation on X. Hence by 3.1, K(T, .X) c S. 0

3.3 says that each B E K(T, 2) is a semigroup algebra, if J? is a Horn theory

over 9. This is, of course, reflected in the diagram in Section 2 and the fact that


the theories 2 mentioned in 2.3, 2.4, 2.5, 2.6, 2.9 are Horn theories. The ,I%” of

2.8 is not a Horn theory as it stands, but this can easily be arranged: each

pseudo-tree algebra is isomorphic to tree&g(&) where ~2 is a pseudo-tree with a

least element; for such &, the last axiom of 2” can be replaced by Vz [t’x (z G

x)-t P(z)], a Horn formula over 2.

3.4. A semigroup generating B(d, E), E a Horn theory over 8. We give a

purely algebraic resp. model-theoretic proof of 3.3, without using the non-trivial

topological equivalence 3.1. Let 2 be a Horn theory over 2 and Se an

Z-structure; we show that B = B(d, 2) is a semigroup algebra by constructing a

semigroup H c B. Write X = X(a, 2). For F a finite subset of A, let

a clopen subset of X. Put

thus H is a subset of B closed under finite intersections and contains 0, = 0, lB = go. H generates B since the elements {U E X: a E r/}, for a E A, are in H. And H is disjunctive: assume that, for F and F,, . . . , F, finite subsets of A,

9F G 9fi u * . * u 9F,

and qF is non-empty. Consider cl(F), the closure of F as defined in 3.2. Since

cl(F) E qFt we have that cl(F) E qF, for some i, i.e., E E cl(F). Now qF G qF; for

U E qF implies F E U, cl(F) G U, c;ls U, U E qF;. One aspect of semigroup algebras is that they show very finitary behaviour.

The following theorem is essentially contained in [2]; a weaker version was

proved in (91.

3.5. Theorem (Bell, Pelant; Heindorf). Zf B is un infinite subalgebra of a semigroup algebra, then B has a countubly infinite homomorphic image.

E.g. no infinite a-complete Boolean algebra embeds into a semigroup algebra.

Theorem 3.5, together with the following results from [14] seems to indicate that

the class of semigroup algebras is the largest model-theoretically defined class all

of whose members have finitary behaviour: every semigroup algebra is a tail

algebra (see Section 2) and every tail algebra is disjunctively generated, i.e., it

has a disjunctive set of generators. Disjunctively generated algebras still show

some finitary behaviour, having countably infinite homomorphic images. On the

other hand, every Boolean algebra is embeddable into (even a retract of) a tail

algebra; thus subalgebras of tail algebras can behave in a quite non-finitary way.

For B = B(d, L’), 2 a Horn theory over 2, a consequence of 3.5 can often be

directly established: for assume there exists a strictly increasing chain (Un),,, in

250 S. Koppelberg

X(& 2). Then U = lJ,,, U, is in X(&, Z), and {Un: n E o} U {U} is a closed subspace of X(Se, Z), giving rise to a countable homomorphic image of B. Similarly for a strictly decreasing chain (Un)new and U = n,,, U,.

Semigroup algebras tend to have nice combinatorial behaviour, as illustrated in the following theorem.

3.6. Theorem (Heindorf [lo]). A ssume B is a semigroup algebra generated by the

semigroup H s B as in 2.10 and that C c B is a chain in B of regular uncountable cardinal@ K. Then there are subchains X G C and Y G H of cardinal@ K such that X is isomorphic to Y or to the converse Linear order Y-‘.

This theorem has been known before for tree algebras (Brenner-Monk, see [5]) and it holds similarly for pseudo-tree algebras (see [14]). For another consequence, see 4.7.

Recall that a topological space X is said to be supercompact if it has a subbase Y for closed sets which is binary, i.e., n Y;, #% if Y, s Y is linked. If, additionally, Y is normal, i.e., if for disjoint S, T E Y there exist S’, T’ E Y such that S fl T’ = 0 = T fl S’ and S’ U T’ =X, then X is normally supercompact. There is an example by van Douwen-van Mill showing that not every semigroup algebra has a supercompact Stone space. But, conversely, a result by van Mill [15] implies the following, as observed by Heindorf.

3.7. Theorem (van Mill; Heindorf). Zf the Stone space of B is normally supercompact, then it admits a multiplication as in 3.1; hence B is a semigroup algebra.

If B = B(&, ,Z’) and 2 is a Horn theory over 2, then inspection of _Z sometimes shows that B has supercompact Stone space. For assume each o E ,Y has the form (3) with n < 1 or (4) with n < 2. Then

Y={{UEX(&,Z):aEU}:aEA}

U{{UEX(~,Z):~.$U}:~EA}

is a clopen subbase of X(&, .Z), since it consists of clopen sets generating

B(.& z), and it is easily seen to be binary. We obtain that the algebras mentioned in Examples 2.3, 2.4, 2.5, 2.8, 2.9 have supercompact Stone spaces.

4. The impact of T on W( T, Z)

Theorem 3.3, or rather its proof, shows how a syntactical property of 2 has a rather strong consequence for the Boolean algebras B(d, Z) -if ,Z is a Horn theory over 2 then B(&, 2:) is a semigroup algebra. The following, admittedly vague, question seems to be quite natural.


Question 1. How are the model-theoretic properties of T, or of T and .Z,

reflected in the class K(T, Z)?

We make some modest contributions to this type of question in the present

section.

4.1. Remark. If ti is an elementary substructure of the _Y-structure 93 then

B(d, _Z) is canonically embedded into B(%, 2) - in fact, we will always identify

B(d, 2) with a subalgebra of B(%?, 2).

A topological explanation for this is as follows. The function

mapping V to V VIA is well-defined by 1.2(a) and onto by 1.2(c); it is clearly

continuous. The dual h of q is a Boolean homomorphism embedding B(&, 2)

into B(B, 2).

We give an algebraic description of h. Denote the ideal I of Fr(A) defined in

1.3(a) by !d and the corresponding ideal of Fr(B) by &. 1.2(b) and & < 3

guarantee that e E Fn(A, 2) is consistent with ti, Z iff it is consistent with %‘, 2;

thus Z, = 1, fl B(&, 2). By 1.3(a), there is a unique embedding of B(&$, 2) into

B(93, 2) mapping a/C (the equivalence class of a modulo 4,) to a/&, for each

free generator a E A E Fr(A). As in the remark following 1.2, it suffices to assume that ti c 3 if all o E 2 are

universal closures of quantifier-free formulas of Z(P).

4.2. Remark. If (&;)iel is an elementary chain of models of T, then ti = lJic, di

is again a model of T, by Tarski’s theorem. Clearly B(z4, Z’) is the union of the

chain (B(~i, X))ie,.

If P’ is a unary predicate not in 9, we denote by Z(P’) the result of replacing, in

each o E 2, P by P’.

4.3. Proposition. The following are equivalent, for arbitrary T and 2: (a) W( T, 2’) contains Boolean algebras of arbitrarily high cardinality. (b) W( T, 2) contains an infinite Boolean algebra. (c) The theory

T U U X(PE) U (3x (P,(x)++lP,(x)): n fm ino} nto

in 9 U {P,, : n E CO}, the P, distinct unary prediates not in 2, is consistent.

Proof. Trivially, (a) implies (b). (b) implies (c) because the Stone space of an

infinite Boolean algebra is infinite. If (c) holds and K is any infinite cardinal, we

obtain an algebra in YQT, E) of cardinality at least K as follows. Let I be a set

252 S. Koppelberg

such that 111 > 2j‘ for all p < K; by (c) the theory

T = T U g 2(e) U (3x (e(x) -14(x)): i fj in Z}

is consistent. Hence there is & L T such that 1X(&, _E)la (I(. If B = B(.&, 2) has

cardinality p < K, then 1X(&, z)l G 2@ < 111, a contradiction. 0

Call the pair (T, 2) non-trivial if it satisfies (a)-(c) in 4.3. For non-trivial

(T, 2), there are Lowenheim-Skolem like effects.

4.4. Proposition. Let (T, _E) be non-trivial and K 3 IZ’l.

(a) Assume %I k T is such that IB(C%, E)( 2 K and that M c B(93,2), (MI SK. Then there is &-c 93 such that IAl = IB(&, 2)( = K and ME B(&, 2). In particular, K(T, 2) contains an algebra of size K.

(b) Assume &k T and o 6 IB(&, 2)l G K. Then there is (82 > .& such that

ICI = IB(%‘, zc)l = K.

Proof. (a) In the notation of 4.1, B(93, E) is generated by the subset {b/Z,: b E B}. So there is ti < 6% such that IAl = K = I{a/Z,: a E A}1 and each element of M

is generated by {a/Z%: a E A}. This ti works for the claim.

(b) Let Z and T, be as in the proof of 4.3; let 99’ be a model of q U eldiag &. The reduct 99 of 9 to 9 satisfies IB(%‘, 2)( > K (as in 4.3) and ti i 6%‘. Choose

% 4 3 as in the proof of part (a) such that ti c %? and IB(%‘, X)1 = K. 0

We have not yet given an example of an interesting class of Boolean algebras

which is not model-theoretically defined. This is easily done by 4.4 and 4.2.

4.5. Corollary. The class of complete Boolean algebras is not model-theoretically dejked; similarly for the classes of o-complete algebras resp. algebras with the countable separation property.

Proof. Denote this class by K and assume K = H(T, 2). By 4.4, there are a

cardinal K and a model d of T such that K~ > K and B = B(Se, 2) has cardinality

K. But then by [12, Theorem 12.11, B cannot be complete. 0

We finally indicate how categoricity or stability of T have strong consequences

on K(T, 2).

4.6. Proposition. Assume 1281 C K, and T is A-categorical for all A 2 K and (T, EC) non-trivial. Then K(T, 2) is A-categorical for all Iz 2 K (i.e., W(T, 2) contains, up to isomorphism, exactly one algebra of size A).


Proof. For A 2 K, let .& be the unique T-model of size A and Bn = B(J&, 2). It

suffices to show that (BL( = A; for then BA is the unique algebra of size A in

W, 2). Applying 4.3 and then, for A instead of K, 4.4(a), we obtain ti 1 T such that

(A( = IB(&, .Z’)I = A. Now ~2 = J?& and we are finished. q

In the two subsequent results, we use the following notation. For ti an

.Z’-structure and ti = (a,, . . . , uk) E Ak, k E w, let, as in 3.4,

q,={UEX(Lzz,~):a I,.., ,UkEU},

an element of B(d, 2) = ClopX(d, 2’). Define a partial orderingc (more

precisely, &) on A“ by

a&6 e qaGq&

The 2k-ary relation C is not necessarily definable in &. But if E is a Horn theory,

then (using the notation of 3.2) we find that

e A (bi E cl@,, . . . ) ak)) i=l

Thus if the k + 1-ary relation “x E cl(x,, . . . , xk)” is definable in &, then so is 5.

This happens in all of our examples 2.3, 2.4, 2.5, 2.6, 2.8, 2.9, 2.10.

4.7. Proposition. Assume T is stable, Lo is definable in &, for all &L T and k E o, and 2 is a Horn theory. Then no algebra in K(T, 2) has an uncountable chain.

Proof. Assume B = B(d, 2’) where d L T and there is an uncountable chain in

B. By 3.3, 3.4, 3.6 there is an uncountable chain Yin

H\{O,} = {qa: a eAk, for some k E o}.

Pick k E w such that Y, = Y fl {qa: a EA’} is infinite; without loss of generality,

there is a sequence (a,),,, in A such that q,,(, s qa, E . . . , i.e., a, c al c . . . . Let

p)(X, y) be an .L!!-formula defining c~, in &. QJ and the sequence (c?,,)~~~~ show

that T is nonstable (see [16, Theorem 2.151). 0

4.8. Proposition. Assume that, for some k E w, ck is definable in T by a formula v(j, y), and there is ti L T and a sequence (an)ntm in Ak such that (q,-,,),,,, is a strictly increasing chain in B(&, E). Then each linear order is embeddable into an algebra in K(T, Z).

254 S. Koppelberg

Proof. Given a linear order (X, <), let c, be a k-tuple of pairwise distinct new

constants, for x E X. The above assumptions guarantee that the theory

T’ = T U { cp(&., EY) A lcp(C,, 2,): x <y in X}

is consistent. Fix a model a’ of T’ and let ci, E Ak be the interpretation of 15, in

&‘. For & the reduct of &’ to 2, the chain {qu;: x E X} in B(&, 2) is isomorphic

to (X, <). 0

Propositions 4.6 through 4.8 suggest the following questions.

Question 2. Are there natural syntactical condition on T, J? implying that, for

each K 2 121, K(T, C) has 2” non-isomorphic members of size 2”?

Question 3. Suppose 2 is countable and T o,-categorical; let (by (4.6) B, be the

unique algebra in M(T, Z), for K 2 wl. For which T does it hold that B, is the

free Boolean algebra over K generators ? In general, how do the algebras B, look

like?

The following example provides some substance to this question.

4.9. Example. Fix a finite field K and let T be the theory of infinite-dimensional

vector spaces over K; T is K-categorical for all K 2 w. Let 2 say that P defines a

K-subspace. Clearly 2 is a Horn theory; for Ju b T and X E M, cl(X), as defined

in 3.2, is the K-subspace of JU generated by X. For each k E CD, the relation

“X E Cl(Xi, . . . ) xk)” is definable in T, by finiteness of K.

For K 2 w, B, = B(.&, 2) (J& the K-vector space of dimension K) is the

unique algebra of size K in bi(T, 2). B, is easily seen to be atomless; moreover,

no ultrafilter of B, is generated (as a filter) by less than K elements, i.e., each

point in Ult(B,) has character K. And by 4.7, B, has no uncountable chain.

B, contains an independent subset Y of size K, e.g. fix a base Z of Ju, over K and let

We proceed to establish a non-trivial property of B, which makes it look quite

similar to E-(K), the free Boolean algebra on K generators-B, has many

relatively complete subalgebras. Namely, given a K-subspace & of J&, we know

from 4.1 that Bd = B(d, 2) is a subalgebra of B, = B(JU,, 2). We sketch a

proof that Bd is relatively complete in B,. This amount to proving (see [12, 8.201) that the natural map

3: X(.&, 2)+X(&, Z),

W -WnA


is open. So let F and G be finite subsets of the underlying set M of .&, with the

aim of showing that the image of the basic clopen subset

N”tlc(F,G)={W~X(~,_2):F~W,Gq14\W}

of X(JuK, .Z) under ~JJ is open, in X(Se, E). Consider the subsets

S = cl(F) f~ A,

T={g-x:g~G,x~cl(F)}nA

of A; they are finite since K is finite. It is not difficult to check that

q[N&=(F, G)]=N&(S, T)

={VEX(&~):S~V, TcA\V},

a clopen subset of X(&, Z).

It follows that for w G A < K and C a subalgebra of B, such that ICI = A, there

is a relatively complete subalgebra D of B, such that C G D and IDI = A. Similarly, B, is the union of an increasing chain (C,),,, of subalgebras such that

]C,l <K, C, is relatively complete in B,, and for limit ordinals (Y < K,

C, = u,<, C,. It follows from a theorem by SEepin [18], [13, 2.8 and 3.41) that B,, = Fr(wl);

trivially, B, = Fr(o) since B, is countable and atomless.

However, B,, + Fr(oJ (and similarly for K 2 02). To see this, choose

K-subspaces V, I& W,, of .A&,, such that dim, V = co,, dim, v, = W, dim, w, =

o, V, c V, and V II WI = (0). Let

and B, the subalgebra of B,, generated by {{W E X(J&,,, 2): y E W}: y E Y}. B, is not relatively complete in B,,-e.g. for a E cl(V U H$)\ Y, the set

{b E B,: b s {W E X(.4&,,,, 2): a E W})

has no greatest element. But if B,, = Fr(02), then an argument as in [18] shows

that there are V, 1/;,, W, as above such that B, is relatively complete in Bw2. Let us finally remark that B,, for K > co, satisfies the countable chain

condition. For if X E B, consists of pairwise disjoint elements and (Xl = wl, pick

a K-subspace ti of JUT such that X E B(,d, 2) and IAl = co,. Now B(sd, 2) = B,, is free, as shown above, and satisfies the countable chain condition, a

contradiction.

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[I] M.G. Bell, The space of complete subgraphs of a graph, Comm. Math. Univ. Carolinae 23 (1982) 525-535.

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256 S. Koppelberg

[3] G. Brenner, Tree algebras, Ph.D. thesis, University of Colarado, 1982.

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[6] C.C. Chang and H.J. Keisler, Model Theory, 3rd edition (North-Holland, Amsterdam, 1990).

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[13] S. Koppelberg, Projective Boolean algebras, in J.D. Monk and R. Bonnet, eds., Handbook of

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[14] S. Koppelberg and J.D. Monk, Pseudo-trees and Boolean algebras, Order 8 (1992) 359-374.

[15] J. van Mill, Supercompactness and Wallman spaces, Math. Centre Tracts 85 (Amsterdam, 1977).

[16] A. Pillay, An Introduction to Stability Theory (Oxford, 1983).

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Boolean Algebras, Vol. III (North-Holland, Amsterdam, 1989) 1097-1165.

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(5) (1976) 155-191.

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A construction of Boolean algebras from first-order structures